Holder continuity and minimal displacement

Let M be a bounded metric space and suppose T : M --> M satisfies for some fixed h > 0, d(T(x), T(y)) less than or equal to max{d(x, y), h} for all x,y is an element of M. It is shown that under this assumption, for certain spaces M there will always exist z is an element of M such that d(z, T(z)) < h. If M is a convex subset of a Banach space, then weak compactness and normal structure suffice. If M is a hyperconvex metric space (in particular, if M is an intersection of closed balls in l(infinity)) then there always exists z is an element of M such that d(z, T(z)) less than or equal to h/2. Mappings of the type considered arise naturally. For example if K is a convex subset of a Banach space and if h,p is an element of (0, 1], then mappings T : K --> K which are Holder continuous in the sense \\T(x) -T(y)\\less than or equal to h\\x-y\\(p), (x, y is an element of K), obviously satisfy the weaker condition \\T(x) -T(y)\\less than or equal to max{h, \\x-y\\}, (x, y is an element of k).